Inna Urazova1 and Ruslan Simanchev1,2
1 Omsk State University, 55a Mira avenue, 644077 Omsk, Russia urazovainn@mail.ru
2 Omsk Scientific Center of SB RAS, 15 Marksa avenue, 644024 Omsk, Russia osiman@rambler.ru
Abstract. This article continues the work [16] in which polyhedral setting of graph approximation problem is provided, support inequalities to polytope are built. In this work NP-hard of the separation problem of k-parachutes relative to M-graph polytope is proved.
Keywords: polytope, facet inequality, separation problem.
1 Introduction
Let Kn = (V, E) be a complete unoriented graph without loops and multipe edges.
Spanning subgraphH ⊂Kn is called M-graph if each of its connected components is a clique or single-vertex graph. We denote a set of all M-graphs inKn throughµ(V).
LetG⊂Kn be some a priori set spanning subgraph. Approximation problem ofG consists in finding M-graph H minimizing the functional
ρG(H) =|EG∪EH| − |EG∩EH| (1) on set µ(V). Here EGand EH are sets of edges of G and H, respectively. In other words, it is necessary to find such a set of pairwise non-overlapping cliques atV which is as far as possible less (in terms of edges) different fromG.
Harary was the first to state graph approximation problem in [10] in 1955. In the 1960s-1970s in a number of works non-trivial classes of graphs on which the problem is polynomially solvable [7, 18] . In 1986 Krivanek and Moravek considered graph ap- proximation problem as a particular case of the tree clusterization and proved that it was NP-hard [11] . Systematic studies of graph approximation problem began in the last decade when the problem was re-discovered under different names (correla- tion clustering; Cluster editing) by different groups of authors [3, 4, 13]. In particular, the NP-hardness of its various options was established [1, 3, 13] and suggested first ap- proximate algorithms with the guaranteed accuracy evaluation [3, 5, 8]. The best of the currently known graph approximation algorithms finds a guaranteed solution by factor of max. 2.5 worse than the optimum one [2, 17].
Copyright c⃝by the paper’s authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
In this work which continues the work [16] we consider polyheldral setting of the graph approximation problem. The polyhedral approach to the solution of extreme com- binatorial problems consists in the correlation of the problem with a special polytope set as a convex hull of incidence vectors of the admissible solutions and, consequently, the use of convex analysis and integer programming means. In this route a special role belongs to the task of polytope description as a set of solutions to the system of linear equations and inequalities. If a full linear description of the polytope is available, the extreme combinatorial problem reduces to the linear programming problem (possibly - with exponential number of constraints) which quite often enables obtaining effective algorithms for the solution thereof. One of the most widely known examples of such a situation is problem of maximum weighted matching [6, 12]. As a rule, for NP-hard problems a full liner description of the polytope is not known (see, e.g., [9]). Never- theless, the availability of a partial description, i.e., classes of valid, support or facet inequalities enables building evaluations of the target function optimum value, develop special cutting plane procedures.
To state the results of this article, we will introduce the following designations and notions. For any graphD⊂KnviaV DandEDwe will denote a set of its vertices and edges, respectively. For an edgee∈E we will also useuvnotation, whereuandvare vertices fromV incident to edge e. For D ⊆Kn andu∈V viaδD(u) we will denote the set of edges of the D incident to vertex u. If D =Kn, then in this notation we will omit indexD. Each set of edgesR⊂E induces inKn some subgraphT in which ET =R andV T is a set of vertices incident to edges from R. If no ambiguity arises, a graph induced by a set of edgesR will be denoted viaR. For subgraphsD, F ⊆Kn we assume
D∪F =ED∪EF, D∩F =ED∩EF.
We will correlate graph Kn with a Euclidean space RE of dimensions n22−n by correlating each edge with a coordinate axis in RE . This space may be considered as a space of column vectors with the components indexed by elements from E. If x∈RE andR⊆E, then we denote linear form ∑
e∈R
xeviax(R). The incidence vector of the arbitrary graphD⊆Kn without isolated vertices is called vectorxD∈REwith componentsxDe = 1 ate∈EDandxDe = 0 ate /∈ED. The latter rule, obviously, sets a one-to-one correspondence between the set of all subgraphs without isolated vertices ofKn and the set of vertices of the unit cube inRE.
A set P ⊂ RE is called a polytope if it is a convex hull of a finite number of points. Linear inequalityaTx≤a0 (a, x∈RE, a̸= 0, a0 ∈R,”T” transposition sign) is called support to P if it is fulfilled for any point fromP and there exists at least one point fromP converting it into an equality. Any inequality support toP generates the set{x∈P|aTx=a0}which is called the face of the polytopeP. Maximum by the inclusion faces of the polytope are called facets. It is clear that a facet is the face and only the face the dimensionality of which is by 1 smaller than the dimensionality of the polytope itself. A support inequality generating a facet will, consequently, be referred to as a facet inequality.
Facet inequalities are present (with the equivalency accuracy) in any linear descrip- tion of the polytope [14, 12]. Besides, they proved to be good as cutting planes for the solution of high dimensionality problems (see, e.g., [15, 9]).
During the development of cutting plane procedures using support inequalities the separation problem goes to the foreground. It consists in the following. What is given is a class of inequalities L support to polytope P and point ¯x ∈ RE. It is required to find an inequality in the class L strictly separating the point ¯xfrom polytope P, or prove that inL such an inequality does not exist. Examples of extreme combinato- rial problems and inequality classes for which the separation problem is polynomially solvable, for example, travelling salesman problem and subtour elimination inequalities [9], the problem of building a multi-processor schedule and inequality class induced by the paths in the precedence graph [15] etc. are known. In this work NP-hardness of the separation problem of k-parachute class inequalities relative to M-graph polytope is proved.
2 M -graph polytope and k-parachutes
We will refer to the set
Pn=conv{xH ∈RE|H ∈µ(V)} as an M-graph polytope. Here ”conv” means ”convex hull”.
In [16] it was prove that (0,1)-vectorx∈REis the vector of incidences ofM-graph if and only if it satisfies the system:
−xuv+xuw+xvw ≤1 xuv−xuw+xvw≤1, xuv+xuw−xvw ≤1
(2)
whereu, v, w∈V are all kinds of sets of three pairwise different vertices,
xuv≥0 for alluv∈E. (3)
Besides, in the same work it was demonstrated that the target function (1) in terms ofRE may be written as
ρG(x) =|EG|+x(EG)¯ −x(EG). (4) Therefore, graph approximation problem may be stated as a problem of integer linear programming (2)–(4).
LetU ={u1, u2, . . . , uk} and W ={v1, v2, . . . , vp} be nonempty subsets of set V, U ∩W = ∅, k ≥1, p≥ 2 . We will designate star in Kn with the centre in vertex ui and arms uivj, j = 1,2, . . . , pthrough Ti, i= 1,2, . . . , k. We will denote the clique on the set of verticesW throughKp. Let us assume thatT =
∪k i=1
Ti. GraphT ∪Kp is called ak-parachute. We will correlate inequality
x(ET)−x(EKp)≤k2+k 2 .
with this graph. In [16] it was proved that this inequality induced byk-parachuteT∪Kp
is support inequality to polytope Pn if and only if p≥k , and facet inequality if and only ifk= 1.
3 Separation Problem
Let L be a class of support inequalities to polytope Pn induced by k-parachutes.
Whereas polytopePn completely lies in the unit cube ofRE, we will state the separa- tion problem stronger than it was done in the Introduction.
Problem A. Let ¯x∈RE and 0≤x¯≤1. DoesLclass contain such an inequality for which ¯x(ET)−x(EK¯ p)>k22+k ?
We need auxiliary fact, which is easily proved by induction.
Lemma 1. If anytedges witht < nare withdrawn from the completen-vertex graph, the remaining graph will contain clique of the order n−t.
Theorem 1. The separation problem for the inequalities induced by k-parachutes rel- ative to the polytope of the graph approximation problem is NP-hard.
Proof. Let us prove that this problem is NP-hard already at k= 1, i.e., atU ={u}. In this case Problem A may be re-stated as follows:
Problem B. Let ¯x ∈ RE and 0 ≤x¯ ≤1. Is there among inequalities of the type
∑p j=1
xuvj −p∑−1
i=1
∑p j=2,j>i
xvivj ≤1 such an inequality which is violated by point ¯x?
It is clear that in this problemuvertex may be fixed. Now at preset point ¯x∈RE, 0≤¯x≤1 and vertexu∈V we will define vectorc∈RE:
ce=
{x¯e, e∈δ(u);
−x¯e, e∈E\δ(u).
We will denote this edge weighted graph viaKn(c, u). It is not difficult to see that Problem B is equivalent to the following problem.
Problem C. Does this edge weighted graph Kn(c, u) contain such clique K that c(EK)>1?
Let us reduce CLIQUE problem stated as follows to Problem C: what is given is graphGand natural numbers <|V G|. Does graphGcontain clique of the order larger thans?
So, let what is given be graphG,|V G|=mand a natural numbers < m. Let us assume thatn=m+ 1 and build a complete graphKn on a set of verticesV G∪ {u} whereuis an added vertex. Let us assign the following weights toKn edges:
ce=
1
s, e∈δ(u);
0, e∈EG;
−1s, e∈EG,¯ where ¯Gis the complement of graphG.
Let us note that if in graphGthere is cliqueKof the order larger thans, then for the cliqueK′ ⊂Kn(c, u) on the set of verticesV K∪ {u}we have c(EK′)>1.
Let us suppose that in graph Kn(c, u) there is clique ¯K ⊂ Kn(c, u) such that c(EK)¯ > 1. Because only edges from δ(u) have positive weights, then necessarily u ∈ VK. It means that clique ¯¯ K may be presented as ¯K = δK¯(u)∪K where K is clique inG∪G. Clique¯ K , in it turn may be represented asK= (K∩G)∪(K∩G).¯ Therefore, ¯K =δK¯(u)∪(K∩G)∪(K∩G). Because edge sets of these three graphs¯ do not overlap pairwise, then
c(EK) =¯ c(δK¯(u)) +c(E(K∩G)) +c(E(K∩G)) =¯
= 1
s|V K| −1
s|E(K∩G)¯ |=1
s(|V K| − |E(K∩G)¯ |)>1.
Hence, particularly, it follows that|V K|>|E(K∩G)¯ |. Let us note that the with- drawal of edges E(K ∩G) from clique¯ K gives exactly K ∩G graph which, first, lies fully in G, and second, by virtue of the lemma, contains clique of the order
|V K| − |E(K∩G)¯ |> s.
The theorem is proved.
4 Conclusion
This article continues the work [16] in which facet inequalities to the graph approxi- mation problem polytope is built. The next step in this direction is the inclusion of the received facet inequalities into polyhedral type algorithms. The effectiveness of this in- clusion depends essentially on the computational difficulties of the separation problem of inequalities. In this work N P-hardness of the separation problem of k-parachutes relative toM-graph polytope is proved. The most natural direction for further research is the development of heuristics to separation problem.
In conclusion, we will announce preliminary results of our numerical experiment.
The aim of the experiment is the evaluation of the use of 1-parachutes in cutting plane and branch-bound algorithms to solve the graph approximation problem. Below are brief results which are as follows. We solve two integer linear programm. The first problem have function (4) as an objective function and have polyhedron (2)–(3) as a relaxation set. In the second problem, we change the polyhedron, adding to the con- straints (2)–(3) all 1-parachute inequalities. This, of course, imposes serious restrictions on the dimension of the problem. To solve this problem we used IBM ILOG CPLEX Optimization Studio package and Intel(R) Celeron(R) CPU N2830 2.16GHz. Solution time was limited to 2 hours. More than a hundred problems with different objective functions of the form (4) for n ≤25 were solved. Table 1 shows the average solving time for the first type and second type problems (columns SR and PR, respectively).
As the table shows, the addition of 1-parachutes greatly reduces the time to solve the problem.
The authors are grateful to A.V. Kononov for useful advice received when working on an article.
Table 1.
n N umber of variablesSR(mean time, sec) PR(mean time, sec)
10 45 0.16 0.23
15 105 0.66 1.99
20 190 46.91 35.35
25 300 6566.36 1464.68
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